10
Discrete-Time
Fourier Series
In this and the next lecture we parallel for discrete time the discussion of the
last three lectures for continuous time. Specifically, we consider the representation of discrete-time signals through a decomposition as a linear combination of complex exponentials. For periodic signals this representation becomes the discrete-time Fourier series, and for aperiodic signals it becomes
the discrete-time Fourier transform.
The motivation for representing discrete-time signals as a linear combination of complex exponentials is identical in both continuous time and discrete time. Complex exponentials are eigenfunctions of linear, time-invariant
systems, and consequently the effect of an LTI system on each of these basic
signals is simply a (complex) amplitude change. Thus with signals decomposed in this way, an LTI system is completely characterized by a spectrum of
scale factors which it applies at each frequency.
In representing discrete-time periodic signals through the Fourier series,
we again use harmonically related complex exponentials with fundamental
frequencies that are integer multiples of the fundamental frequency of the periodic sequence to be represented. However, as we discussed in Lecture 2, an
important distinction between continuous-time and discrete-time complex
exponentials is that in the discrete-time case, they are unique only as the frequency variable spans a range of 27r. Beyond that, we simply see the same
complex exponentials repeated over and over. Consequently, when we consider representing a periodic sequence with period N as a linear combination
of complex exponentials of the form efkOn with o = 27r/N, there are only N
distinct complex exponentials of this type available to use, i.e., efoOn is periodic in k with period N. (Of course, it is also periodic in n with period N.) In many
ways, this simplifies the analysis since for discrete time the representation involves only N Fourier series coefficients, and thus determining the coefficients from the sequence corresponds to solving N equations in N unknowns.
The resulting analysis equation is a summation very similar in form to the synthesis equation and suggests a strong duality between the analysis and synthesis equations for the discrete-time Fourier transform. Because the basic
10-1
Signals and Systems
10-2
complex exponentials repeat periodically in frequency, two alternative interpretations arise for the behavior of the Fourier series coefficients. One interpretation is that there are only N coefficients. The second is that the sequence
representing the Fourier series coefficients can run on indefinitely but repeats periodically. Both interpretations, of course, are equivalent because in
either case there are only N unique Fourier series coefficients. Partly to retain
a duality between a periodic sequence and the sequence representing its
Fourier series coefficients, it is typically preferable to think of the Fourier series coefficients as a periodic sequence with period N, that is, the same period
as the time sequence x(n). This periodicity is illustrated in this lecture through
several examples.
Partly in anticipation of the fact that we will want to follow an approach
similar to that used in the continuous-time case for a Fourier decomposition
of aperiodic signals, it is useful to represent the Fourier series coefficients as
samples of an envelope. This envelope is determined by the behavior of the
sequence over one period but is not dependent on the specific value of the period. As the period of the sequence increases, with the nonzero content in the
period remaining the same, the Fourier series coefficients are samples of the
same envelope function with increasingly finer spacing along the frequency
axis (specifically, a spacing of 2ir/N where N is the period). Consequently, as
the period approaches infinity, this envelope function corresponds to a Fourier representation of the aperiodic signal corresponding to one period. This is,
then, the Fourier transform of the aperiodic signal.
The discrete-time Fourier transform developed as we have just described
corresponds to a decomposition of an aperiodic signal as a linear combination of a continuum of complex exponentials. The synthesis equation is then
the limiting form of the Fourier series sum, specifically an integral. The analysis equation is the same one we used previously in obtaining the envelope of
the Fourier series coefficients. Here we see that while there was a duality in
the expressions between the discrete-time Fourier series analysis and synthesis equations, the duality is lost in the discrete-time Fourier transform since
the synthesis equation is now an integral and the analysis equation a summation. This represents one difference between the discrete-time Fourier transform and the continuous-time Fourier transform. Another important difference is that the discrete-time Fourier transform is always a periodic function
of frequency. Consequently, it is completely defined by its behavior over a frequency range of 27r in contrast to the continuous-time Fourier transform,
which extends over an infinite frequency range.
Suggested Reading
Section 5.0, Introduction, pages 291-293
Section 5.1, The Response of Discrete-Time LTI Systems to Complex Exponentials, pages 293-294
Section 5.2, Representation of Periodic Signals: The Discrete-Time Fourier
Series, pages 294-306
Section 5.3, Representation of Aperiodic Signals: The Discrete-Time Fourier
Transform, pages 306-314
Section 5.4, Periodic Signals and the Discrete-Time Fourier Transform, pages
314-321
Discrete-Time Fourier Series
10-3
MARKERBOARD
10.11
Note that dt should be added at the end of the last equation in column 1.
MARKERBOARD
10.2
Signals and Systems
10-4
Example 5.2:
x[n]
= 1 + sin £ 0 n + 3 cos 2 0 n
+ cos (2E2 0 n +
Re ak
TRANSPARENCY
10.1
Example of the
Fourier series
coefficients for a
discrete-time periodic
signal.
)
iak
3
"1-0/
k
N
0
N
k
N
0
N
<%ak
ir/2
*-IIi
I:,
a1
- -
TRANSPARENCY
10.2
Comparison of the
Fourier series
coefficients for a
discrete-time periodic
square wave and a
continuous-time
periodic square wave.
0-TI
--I
-
k
N
Discrete-Time Fourier Series
10-5
TRANSPARENCY
10.3
Illustration of the
discrete-time Fourier
series coefficients as
samples of an envelope. Transparencies
10.3-10.5 demonstrate
that as the period
increases, the
envelope remains the
same and the samples
representing the
Fourier series
coefficients become
more closely spaced.
Na.
Here, N = 10.
TRANSPARENCY
10.4
N = 20.
Nao
Envelope:
0 27r
20
sin[(2N 1 + 1) 92/2]
sin (W/2)
-F.1N
- 20
Signals and Systems
10-6
TRANSPARENCY
10.5
N = 40.
Nao
Envelope:
Afr~r
sin[(2N 1 + 1) 92/2]
sin (92/2)
.
N - 40
2i
1. x(t) APERIODIC
TRANSPARENCY
10.6
A review of the
approach to
developing a Fourier
representation for
aperiodic signals.
- construct periodic signal X(t) for
which one period isx(t)
- x(t) has a Fourier series
-
as period of x(t) increases,
x(t) -- x(t) and Fourier series of
x(t)
Fourier Transform of x(t)
Discrete-Time Fourier Series
10-7
1. x[n] APERIODIC
TRANSPARENCY
10.7
A summary of the
approach to be used to
obtain a Fourier
representation of
discrete-time
aperiodic signals.
- construct periodic signal x[n]for
which one period isx[n]
- x[n]has a Fourier series
- as period of x[n] increases,
i I[n] -.-x[n] and Fourier series of
x[n]
--
Fourier Transform of x[n]
FOURIER REPRESENTATION OF APERIODIC SIGNALS
TRANSPARENCY
10.8
Representation of an
aperiodic signal as the
limiting form of a
periodic signal with
the period increasing.
inJ
...
*jaaTTT??
nTTITTT.
?TTTATTTTTTL
??T????TTr
-N
-N1
0
x[n] = x[n]
As N -+oo
- let N
-+
N1
In I < 2
x[n]
o x[n]
oo to represent x[n]
- use Fourier series to represent x[n]
N
n
Signals and Systems
10-8
Example 5.3:
TRANSPARENCY
10.9
Transparencies 10.910.11 illustrate how
the Fourier series
coefficients for a
periodic signal
approach the
continuous envelope
function as the period
increases. Here, N =
10. [Example 5.3 from
the text.]
60 9
U
-NI?.. - N0
-1
-N
-N,
0
- W
N
N,
n
Nao
/
Envelope:
/
sin[ (2N 1 + 1)
N = 10
/2]
sin (W/2)
Nai
0
2r
10
Example 5.3:
TRANSPARENCY
10.10
N = 20. [Example 5.3
from the text.]
I*
-N
A'.ft
-
-
-N,
1
-.--
n
N
Nao
/
/ -'TNNai
0 2w
20
Envelope:
sin( [(2N 1 + 1) R/2]
sin (fl/2)
N - 20
I
Discrete-Time Fourier Series
10-9
-N
Na 1
/
-11 N1
N11?...
-N1
N
0
Envelope:
/
in]
=
N1
1XWkn)
I
k=<N>
e jkE2Gn go
n /
n=-N/2
N - 40
TRANSPARENCY
10.12
n= N/2
As N
10.11
N = 40. [Example 5.3
from the text.]
sin [(2NI + 1) 2/2)
sin (U/2)
X(kg2o) = N ak =
TRANSPARENCY
x[n] e-jk on
oo
Limiting form of the
Fourier series as the
period approaches
infinity. [The upper
limit in the summation in the second
equation should be
n = (N/2) - 1.]
Fourier Transform:
x[n]
X(92)
=
=
2'r 2ir
+oo
X(92) ein" d92
x[n] e-in"
n=-oc
Signals and Systems
10-10
DISCRETE-TIME FOURIER TRANSFORM
TRANSPARENCY
10.13
The analysis and
synthesis equations
for the discrete-time
Fourier transform. [As
corrected here, x[n],
not x(t), has Fourier
transform X(D).]
-
x [n]
2
X(E2) en"d
synthesis
x[n]
analysis
7
+o
X (2)
=
n=-oo
x[n] <
X(2)
=
=
XG
e X(2)
+ j Im I X(&)4
|X(S2)1 ei"(2)
oI
II
x [n]
TRANSPARENCY
10.14
The discrete-time
Fourier transform for
a rectangular pulse.
-2
0
n
2
X (W)
-27r
27r
1-2
Discrete-Time Fourier Series
10-11
x[n]= an u[n]
O< a < 1
TRANSPARENCY
10.15
The discrete-time
Fourier transform for
an exponential
sequence.
+00
X(W)
an u[n]e-inn
n=-00
an e-j2n
n= 0
1
1 -a
x[n]
2
anu[n]
0<a<1
X(2)
1a e-iW
1X()| I
TRANSPARENCY
10.16
-2r
-7
27r
0
, X(W)
tan~ (a/v'Y 2)
92
Illustration of the
magnitude and phase
of the discrete-time
Fourier transform for
an exponential
sequence. [Note that a
is real.]
Signals and Systems
10-12
2. R(t) PERIODIC, x(t) REPRESENTS ONE PERIOD
TRANSPARENCY
10.17
A review of some
relationships for the
Fourier transform
associated with
periodic signals.
Fourier series coefficients of 2(t)
times samples of Fourier.
= (1/T)
transform of x(t)
3. 2*(t) PERIODIC
-Fourier transform of x(t) defined as
impulse train:
+oo
X(W)
27rak 6 (o - ko)
=
k=I-oo0
2. '[n] PERIODIC, x[n] REPRESENTS ONE PERIOD
- Fourier series coefficients of 2[n]
TRANSPARENCY
10.18
A summary of some
relationships for the
Fourier transform
associated with
periodic sequences.
= (1/ N ) times samples of Fourier
transform of x [n]
3. X[n] PERIODIC
-Fourier transform of x[n]defined as
impulse train:
X(n) =
27rak 5 (2 - k20 )
Discrete-Time Fourier Series
10-13
Fourier series coefficients equal
1
- times samples of Fourier
transform of one period
TRANSPARENCY
10.19
The relationship
between the Fourier
series coefficients of a
periodic signal and the
Fourier transform of
one period.
x~nJ
-N
??TYATTTT?
flT??,??TTTT
-N,
x [n]
0
6 one period of
x [n]
N1
N
n
x[n]
o ak
x [n]
0X (9)
k
N
2irk
N
TRANSPARENCY
10.20
Illustration of the
relationship in
Transparency 10.19.
Signals and Systems
10-14
2. *[n] PERIODIC , x[n] REPRESENTS ONE PERIOD
- Fourier series coefficients of *[n]
TRANSPARENCY
10.21
A summary of some
= (1/ N ) times samples of Fourier
relationships for the
Fourier transform
associated with
periodic sequences.
[Transparency 10.18
repeated]
transform of x [n]
3. #x[n] PERIODIC
-Fourier transform of xIn]defined as
impulse train:
X (9) =
..1111I
27rak 5(2 - k20 )
11111
N1 0
N1
~T1111"
__
TRANSPARENCY
10.22
Illustration of the
Fourier series
coefficients and the
Fourier transform for
a periodic square
wave.
0
1
0
20
-11111
n
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Professor Alan V. Oppenheim
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